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# Max product of sums that make 100

## Question

What's the maximum product of any whole number of real numbers whose sum is 100? What are they?

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What's the maximum product of any whole number of real numbers whose sum is 100? What are they?

They should be the same numbers, so 100/n is the form and (100/n)n is the solution.

I get n = 37

Product = 9 474 061 716 781 820

Interestingly, n = 25 and n = 50 give the same product, namely 1 125 899 906 842 624.

If n need not be integral, n =100/e gives 9 479 842 689 868 740

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Is it infinity? If I take the 3 numbers 2a+100, -a, -a then the sum is 100 but the product is (2a+100)a^2 which is unbounded as a->infinity.

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36 times e i.e. e^36, multiplied by e^(100-36*e)

The product would be = e^100/e

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